STABILIZATION BY A DIAGONAL MATRIX
C. S. BALLANTINE
Abstract.
In this paper it is shown that, given a complex
square matrix A all of whose leading principal minors are nonzero,
there is a diagonal matrix D such that the product DA of the two
matrices has all its characteristic roots positive and simple. This
result is already known for real A, but two new proofs for this case
are given here.
1. The real case. A theorem proved by Fisher and Fuller [2] is an
obvious consequence of the following result (Theorem 1), which in
turn is the real case of our Theorem 2 below.
Theorem
1 (Fisher,
Fuller).
Let A be an nXn real matrix all of
whose leading principal minors are positive. Then there is annXn
positive diagonal matrix D such that all the roots of DA are positive and
simple.
We shall give here two proofs of Theorem 1, both of them simpler
than the proof in [2]. Our first proof is the shorter of the two, but is
less constructive since it makes use of the continuity of the roots (as
functions of the matrix entries). Our second proof gives explicit (and
relatively simple) estimates for the entries of D in terms of the entries
of A.
First proof of Theorem
1. Here we use induction on n. For « = 1
the result is trivial, so suppose that n 5: 2 and that the result holds for
matrices of order n — 1. Let A be an «X« real matrix all of whose
lpm's (leading principal minors) are positive and let ^4i be its leading
principal submatrix
of order n— 1. Then all the lpm's of A\ are positive, so by our induction assertion there is a positive diagonal matrix
D\ of order n— 1 such that all roots of D1A1 are positive and simple.
Let d be a real number to be determined later (but treated as a variable for the present). Let A be partitioned as follows:
U3
Ad
Received by the editors September 12, 1969.
AMS Subject Classifications. Primary 1555, 1525; Secondary 2660, 2655, 6540.
Key Words and Phrases. Diagonal matrix, stable matrix, leading principal minors,
characteristic
roots, positive simple roots, continuity of roots, separation of roots,
parallelotope, continuous mapping.
728
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stabilization
(where A a is 1X1).
on d) by conformable
by a DIAGONALMATRIX
Define an nXn
diagonal
matrix
729
D (depending
partition:
Lo dj
Let DA =M(d),
where now we emphasize
TDiAi
"(0) = [o
the dependence
on d. Then
DiAi]
o]'
so the nonzero roots of M(0) are just those of D1A1, hence are positive and simple (and there are exactly n— 1 of them). M(0) also has
a simple root at zero. Thus for all sufficiently small d>0 the real
parts of the roots of M(d) are (still) n distinct real numbers at least
n —1 of which are positive. (This follows from the fact that the roots
of M(d) are continuous functions of d.) Choose some such d. Then the
roots of M(d) are all real and simple (since nonreal roots must occur
in conjugate pairs) and at least n — 1 of them are positive. But the
determinant of M(d) is positive since those of A and D are, so in fact
all n roots of M(d) are positive. This concludes the proof of the induction step and hence of Theorem 1.
Remark 1. This same kind of argument can be used to prove that,
when all the 1pm's of A are positive and a sign pattern is given for
w-tuples of real numbers ordered by their absolute values, there is a
real diagonal matrix D of the prescribed sign pattern such that the
roots of DA also have the prescribed sign pattern. Also, when A is an
nXn complex matrix all of whose lpm's are nonzero and also an open
sector containing the positive real axis is prescribed,
this same kind
of argument yields a complex nXn diagonal matrix D such that all
the roots of DA lie in the prescribed sector (in fact, if all the lpm's of
A are positive then D can be chosen positive).
For our second proof of Theorem 1 we shall use the following fact
about real polynomials.
Fact 1. Let n be an integer ^ 2 and let Co, c\, • ■ • , cn be positive
real numbers satisfying all the following inequalities:
2
2
2
4c0c2 < ci, 4cic3 < c2, ■ • • , 4c„_2Cre < c„-i.
Let Xk= (ck+\/ck-i)in for k = 1, 2, ■ • • , n— 1. Then all the roots of the
polynomial
f(x)
= c0xn - ax"-1
+ c2xn~2 -•••+(-
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l)nc„
730
C. S. BALLANTINE
[August
are real (hence positive) and simple, and they are separated by the
n— 1 numbers x\, x2, • • • , x„_i.
Proof. (This result is probably known, is perhaps even classical,
but it is not standard for «S^ 3, so we shall give a short proof here for
the sake of completeness.)
Let x0 = ci/c0 and x„ = c„/c„_i. Then
xo>xi>x2>
• • • >x„_i>x„;
in fact,
\co /
\co /
\C\ /
>£)>
\Ci )
\c2 /
>(~)'
holds by hypothesis, and hence
Xq >
Xi >
c2/ci
>
x2 >
c3/c2 >
• ■■ > Xn_i >
Thus it suffices to show that (-l)*/(x*)>0
x„.
for k = 0, 1, 2, • • • , n.
From the last chain of inequalities we have that cyx*—c/+i is
(1) positive if Q^k<j^n
—1 (and in other cases which we shall
not need here),
(2) negative if 1 ^j'+l <k^n, and
(3) zero if 0 = k=j or j-\-l = k = n. Thus we have
f(x0)
= x0
(c0x0 — Ci) + x0
(c2x0 — c%) +
• • • > 0,
2
(~
!)"/(*•>)
=
(Cn — Cn-lXn)
+
Xn(cn~2
~
C„-SX„)
+
■• • > 0
since n §: 2 and the odd term at the end (when n is even) is positive
in each case.
To handle the x* for which l^k<n,
we write
f(x)
= xn~k+ig(x)
+
(-
l)*x"-*-1(-
Ck-ix2 + ckx -
Ck+i) + h(x),
where we have put
g(x)
= c0x*-2 -
*(*)
=
(ck+2xn-k-2
cix*-3 + C2x*-4 -...+(-
Ck+,xn~k-3+
l)*-2a_2,
•••+(-
\y-k-2cn)(-
1)*+2.
We first show that (-l)*g(x*) ^0 and (-\)kh(xk) ^0. Namely,
(— l)kg(xk)
= (ci-?. — ck-3xk) + xk(ck-i — ck-x,xk) +
(— l)kh(xk)
= xk
(ck+2xk — ck+3) +
xk
(ck+ixk
■ • • ^ 0,
— Ck+s) +
■■■^
0.
Thus it remains only to show that —ck-ixl+ckXk —Ck+i is positive.
But this is a routine consequence of our hypothesis that cl>4c*_i£*+i
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1970]
STABILIZATION BY A DIAGONAL MATRIX
and our definition
of xk. Thus, as asserted,
731
(—1 )*/(#*) >0 for k = 0, 1,
2, • • • , n, and Fact 1 is proved.
Second proof of Theorem 1. We first assume that all lpm's of A
are 1. For k = 1, 2, ■ • • , n let qk be the sum of the absolute values of
the wowleading principal minors of order k in A (thus qn = 0). Choose
positive real numbers d\, d2, • ■ ■ , dn so that
2dk+iqk < dk and
36dk+i < dk
for k = 1, 2, • • • , n— 1. (One can choose dj arbitrarily
>0 for one/
and choose the other dk recursively, going outward from k=j). Let
D=d\ag(di,
d2, ■ ■ ■ , dn), and define c0 ( = 1), C\, c%,• • • , cn by
det (xl — DA)
-
c0xn — cxxn~x + c2x"~2 —••■+(—
l)"c„.
Thus by Fact 1, in order to show that the roots of DA are positive
and simple, it suffices to show that Ci, c2, • ■ • , cn are all positive and
that c£> 4c*_iCfc+ifor k = l, 2, •••,»
—1. This we do as follows.
For k = 0, 1, 2, • • • , n we can write ck = pk+Rk, where pk = did2
■ ■ • dk (hence £o = l) and Rk is the sum of the nonleading
kXk minors in DA (hence R0 = 0 = Rn).
Now, each term in P* is of the form
(*)
principal
dhdh ■ ■ ■djtm,
where m is a nonleading
principal
kXk minor of A and ji<ji<
<jk and k<jk. Since di>d2> ■ ■ ■ >dn>0, the absolute
the term (*) is therefore less than or equal to
■• •
value of
did2 • ■ ■dk-idk+i| m| = (dk+i/dk)pk \m\.
By the Triangle
Inequality
and the definition
of qk we thus have
(for l^fc^w-l)
| Rk | ^
(dk+i/dk)pkqk
< \pk,
the last inequality coming from the way we chose dk and dk+i. Thus
Ck= pk+Rk satisfies pk/2<ck<3pk/2
(for l^k^n
— 1, but also for
k = 0 and for k=n), so
2
2
ck > \pk = l(dk/dk+i)pk+ipk-i
> 9pk+ipk~i > ick+iCk-i
for £ = 1, 2, • • • , n —1. This completes
the proof for the case where
all the lpm's of A are 1.
Returning now to the general case, where the lpm's of A are not
necessarily all 1 (but are all positive), we let mk be the kXk 1pm of A
(hence m0=l) and let
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732
C. S. BALLANTINE
E = diag
(nto/mi,
mi/m2,
[August
• • • , tnn-x/mn).
Then all the lpm's of EA are 1, so we can now apply the proof for
that case and get the required matrix D=diag(di,
■ ■ ■ , d„) for this
general case by choosing di, ■ ■ ■ , dn so that
2
2dk+itnk+imk-iqk
< dkmk
2
and
0 < 36dk+imk+imk-i
< dknik
iork = l,2, ■ ■ ■ , m—1, where now g* is the sum of the absolute values
of the nonleading principal kXk minors of EA. This completes our
second proof of Theorem 1.
2. The complex case. Here we adapt our second proof of Theorem
1 to yield a proof of the complex case (Theorem 2 below). However,
the rest of this latter proof is not constructive,
depending as it does
on the following result from algebraic topology. (This result is a spe-
cial case of [l, Lemma 2, p. 232].)
Fact 2. Let P and Q be n-parallelotopes
in Rn which are parallel
to each other, and let / be a continuous mapping of P into R" such
that/ takes each hyperface of P into the closed supporting half space
of Q at the corresponding
hyperface of Q. Then f(P) includes Q.
Using Fact 2, we can now prove the next theorem, which is the
main result of this section.
Theorem
principal
2. Let A be an nXn
minors
are nonzero.
complex matrix all of whose leading
Then there is an nXn
complex
diagonal
matrix D such that all the roots of DA are positive and simple.
Proof. We follow the lines of our second proof of Theorem 1 where
we can. Without loss of generality we may assume all lpm's of A are
1. For k = 1, 2, • • • , n, again let qk be the sum of the absolute values
of the kXk nonleading principal minors of A. Choose positive real
numbers n, r%, • • • , rn so that
2rk+iqk < rk
and
36ri+1 < rk
for k = 1, 2, • • • , «—1. Now let dk = rk exp i0k for k = l, 2, • ■ • , n,
where 0\, ■ ■ ■ , 0n are real numbers
treated for now as variables. Let
yet to be determined
and are
D = diag (du d2, ■ ■ ■ , dn),
det [pel - DA) = c0x" - dx""1 +•••
(co=l),
as before. Here we define new "variables"
to 6i, ■ ■ • , 0n by means
of the linear
transformation
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+ (-
l)"c„
<fr, • • ■ , <j>„,related
1970]
STABILIZATION BY A DIAGONAL MATRIX
4>k= e1 + e2-\- ■ ■ ■ +ek
6k = <t>k
— 4>k-i
for
for
733
k = 1,2, •••,»,
k = 2, ■ ■ ■ ,n,
61 = fo,
which is one-one of P" onto P".
As before, we write pk = did2 ■ • ■dk and c*=£*-)-P*;
hence, putting
0o = 0, we have
pk = rxr2 ■ ■ ■ rk exp i(Bi + 02 +
• • • + 6k) = rxr2 ■ • • rk exp i<pk
for jfe= 0, 1, 2, • • • , n. Again we find that
I RkI < \pk |/2, I pk|/2 < I ckI < 3 I Pk|/2
for 0 ^ &g »,
and
\ck|2 > 4 I Ck-iI■I c*+iI
for 1 £ * £ » - 1.
Thus by Fact 1 our proof will be complete when we have shown that
we can choose <pi,<p2, ■ ■ ■ , <t>nas real numbers such that Ci, c2, ■ ■ ■ , cn
are all positive.
To show that we can do this, let 1 ^k ^n. For </>&
= j7r, pk=irir2 ■ • •
rk is positive imaginary, so
Im c*= Im pk+lm
Rk= | pk\+Im
Rk^ \ pk | — |P* | >h \ pk\,
and likewise, for <j>k=—§T, Im ck< —\\pk\ ■ Thus we have a continuous mapping
(4>ufc, • • ■ , <t>n)—»(Im ci, Im c2, ■ ■ • , Im c„)
from the
w-cube)
rectangular
w-parallelotope
— §t = 0* = 5T»
(which
here
is actually
an
k = 1, 2, • • • , »,
into real «-space, and this mapping satisfies the hypotheses
relative to the rectangular w-parallelotope
I Im ckI ^ i I pkI = fr^ • • • rk,
of Fact 2
k = 1, 2, • ■• , ».
Thus the range of this mapping includes the latter parallelotope
and
in particular contains the origin. Therefore we can choose <pi, ■ ■ ■ , </>„
all in the interval [ —%w, \ir] so as to yield
(Im ci, ■ • • , Im cn) = (0, • • • , 0)
and hence for this choice of <j>i,■ • ■ , <j>nwe have a, ■ ■ ■ , cn all real.
It is evident geometrically
(and routine to show analytically)
that
Ckcannot be ^0 for —^ir^fa^Jtt,
so in fact a, ■ ■ ■ , cn are all posiLicense or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
734
c. s. ballantine
tive for the above choice of 4>i, ■ ■ ■ , <j>n.This completes
the proof of
Theorem 2.
Acknowledgment.
The author wishes to thank his colleague Dr.
D. H. Carlson and the referee for helpful suggestions concerning this
paper, and particularly
wishes to thank the referee for the reference
[l] below.
References
1. K. Fan,
Topological proofs for certain
theorems on matrices with non-negative
elements, Monatsh. Math. 62 (1958), 219-237. MR 20 #2354.
2. M. E. Fisher and A. T. Fuller, On the stabilization
of matrices and the convergence
of linear iterative processes, Proc. Cambridge Philos. Soc. 54 (1958), 417-425. MR 20
#2086.
Oregon State University,
Corvallis,
Oregon 97331
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